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Introduction Mixing Axions and Large Decay Constants Open issues
Adventures with Stringy Axions
Wieland Staessensbased on (1503.01015, 1503.02965 [hep-th]) + work in progress
with G. Shiu (& F. Ye)
Instituto de Fısica TeoricaUAM/CSIC
Madrid
Iberian Strings 2016,
IFT-UAM/CSIC, 28 January 2016
![Page 2: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/2.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• BICEP2 (2014) ⇒ illusion of large field inflation
Lyth (1996): tensor pert. with r > 0.01single field⇔ ∆φ
∣∣inf> MPl
? Shift symmetry of axions → dim 6 operators slow roll: |η| 1 X
? Nonperturbative effects Vinf ∼ Λ4[1− cos a
f
]⇒ natural inflation Freese-Frieman-Olinto (1990) with f > MPl for slow roll
• Entering String Theory ∼ UV complete theory with quantum gravity⇒ realising f > MPl upon dimensional reduction to 4d
? plethora of axions + local origin of shift symmetries
Banks-Dixon (1988), Kallosh-Linde-Linde-Susskind (1995), Banks-Seiberg (2010), . . .
? But ∃ no-go theorems and arguments forbidding f > MPl (1 axion)
Banks-Dine-Fox-Gorbatov (’03), Svrcek-Witten (’06), Arkhani-Hamed-Motl-Nicolis-Vafa (’06), Conlon-Krippendorf (’16)
; multiple axions: N-flation (2005), aligned natural inflation (2004)
? ∃ no-go theorems for multiple axions?? (Generalized WGC)
Montero-Uranga-Valenzuela (2015), Brown-Cottrell-Shiu-Soler(2015), Heidenreich-Reece-Rudelius (2014/15)
![Page 3: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/3.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• BICEP2 (2014) ⇒ illusion of large field inflation
Lyth (1996): tensor pert. with r > 0.01single field⇔ ∆φ
∣∣inf> MPl
? Shift symmetry of axions → dim 6 operators slow roll: |η| 1 X
? Nonperturbative effects Vinf ∼ Λ4[1− cos a
f
]⇒ natural inflation Freese-Frieman-Olinto (1990) with f > MPl for slow roll
• Entering String Theory ∼ UV complete theory with quantum gravity⇒ realising f > MPl upon dimensional reduction to 4d
? plethora of axions + local origin of shift symmetries
Banks-Dixon (1988), Kallosh-Linde-Linde-Susskind (1995), Banks-Seiberg (2010), . . .
? But ∃ no-go theorems and arguments forbidding f > MPl (1 axion)
Banks-Dine-Fox-Gorbatov (’03), Svrcek-Witten (’06), Arkhani-Hamed-Motl-Nicolis-Vafa (’06), Conlon-Krippendorf (’16)
; multiple axions: N-flation (2005), aligned natural inflation (2004)
? ∃ no-go theorems for multiple axions?? (Generalized WGC)
Montero-Uranga-Valenzuela (2015), Brown-Cottrell-Shiu-Soler(2015), Heidenreich-Reece-Rudelius (2014/15)
![Page 4: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/4.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• BICEP2 (2014) ⇒ illusion of large field inflation
Lyth (1996): tensor pert. with r > 0.01single field⇔ ∆φ
∣∣inf> MPl
? Shift symmetry of axions → dim 6 operators slow roll: |η| 1 X
? Nonperturbative effects Vinf ∼ Λ4[1− cos a
f
]⇒ natural inflation Freese-Frieman-Olinto (1990) with f > MPl for slow roll
• Entering String Theory ∼ UV complete theory with quantum gravity⇒ realising f > MPl upon dimensional reduction to 4d
? plethora of axions + local origin of shift symmetries
Banks-Dixon (1988), Kallosh-Linde-Linde-Susskind (1995), Banks-Seiberg (2010), . . .
? But ∃ no-go theorems and arguments forbidding f > MPl (1 axion)
Banks-Dine-Fox-Gorbatov (’03), Svrcek-Witten (’06), Arkhani-Hamed-Motl-Nicolis-Vafa (’06), Conlon-Krippendorf (’16)
; multiple axions: N-flation (2005), aligned natural inflation (2004)
? ∃ no-go theorems for multiple axions?? (Generalized WGC)
Montero-Uranga-Valenzuela (2015), Brown-Cottrell-Shiu-Soler(2015), Heidenreich-Reece-Rudelius (2014/15)
![Page 5: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/5.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• BICEP2 (2014) ⇒ illusion of large field inflation
Lyth (1996): tensor pert. with r > 0.01single field⇔ ∆φ
∣∣inf> MPl
? Shift symmetry of axions → dim 6 operators slow roll: |η| 1 X
? Nonperturbative effects Vinf ∼ Λ4[1− cos a
f
]⇒ natural inflation Freese-Frieman-Olinto (1990) with f > MPl for slow roll
• Entering String Theory ∼ UV complete theory with quantum gravity⇒ realising f > MPl upon dimensional reduction to 4d
? plethora of axions + local origin of shift symmetries
Banks-Dixon (1988), Kallosh-Linde-Linde-Susskind (1995), Banks-Seiberg (2010), . . .
? But ∃ no-go theorems and arguments forbidding f > MPl (1 axion)
Banks-Dine-Fox-Gorbatov (’03), Svrcek-Witten (’06), Arkhani-Hamed-Motl-Nicolis-Vafa (’06), Conlon-Krippendorf (’16)
; multiple axions: N-flation (2005), aligned natural inflation (2004)
? ∃ no-go theorems for multiple axions?? (Generalized WGC)
Montero-Uranga-Valenzuela (2015), Brown-Cottrell-Shiu-Soler(2015), Heidenreich-Reece-Rudelius (2014/15)
![Page 6: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/6.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• axion monodromy: remains untouched by WGCSilverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008)
Marchesano-Shiu-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Blumenhagen-Herschmann-Plauschinn (2014), Kaloper-Lawrence-Sorbo (2008-2014)
• For N axions & M(≥ N) instantons definition of feff ambiguous; consider the diameter of axion moduli spaceBachlechner-Long-McAllister (2014/15), Junghans (2015)
• Presence of other (heavy) fields affects inflation process⇒ 0.8MPl < feff < MPl consistentwith dataAchucarro-Atal-Welling (2015)
Achucarro-Atal-Kawasaki-Takahashi (2015)
⇒ Understanding of axion EFTindispensable
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Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• axion monodromy: remains untouched by WGCSilverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008)
Marchesano-Shiu-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Blumenhagen-Herschmann-Plauschinn (2014), Kaloper-Lawrence-Sorbo (2008-2014)
• For N axions & M(≥ N) instantons definition of feff ambiguous; consider the diameter of axion moduli spaceBachlechner-Long-McAllister (2014/15), Junghans (2015)
• Presence of other (heavy) fields affects inflation process⇒ 0.8MPl < feff < MPl consistentwith dataAchucarro-Atal-Welling (2015)
Achucarro-Atal-Kawasaki-Takahashi (2015)
⇒ Understanding of axion EFTindispensable
![Page 8: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/8.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• axion monodromy: remains untouched by WGCSilverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008)
Marchesano-Shiu-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Blumenhagen-Herschmann-Plauschinn (2014), Kaloper-Lawrence-Sorbo (2008-2014)
• For N axions & M(≥ N) instantons definition of feff ambiguous; consider the diameter of axion moduli spaceBachlechner-Long-McAllister (2014/15), Junghans (2015)
• Presence of other (heavy) fields affects inflation process⇒ 0.8MPl < feff < MPl consistentwith dataAchucarro-Atal-Welling (2015)
Achucarro-Atal-Kawasaki-Takahashi (2015)
⇒ Understanding of axion EFTindispensable
![Page 9: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/9.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions
• axion monodromy: remains untouched by WGCSilverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008)
Marchesano-Shiu-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Blumenhagen-Herschmann-Plauschinn (2014), Kaloper-Lawrence-Sorbo (2008-2014)
• For N axions & M(≥ N) instantons definition of feff ambiguous; consider the diameter of axion moduli spaceBachlechner-Long-McAllister (2014/15), Junghans (2015)
• Presence of other (heavy) fields affects inflation process⇒ 0.8MPl < feff < MPl consistentwith dataAchucarro-Atal-Welling (2015)
Achucarro-Atal-Kawasaki-Takahashi (2015)
⇒ Understanding of axion EFTindispensable
![Page 10: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/10.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
Axions & String Theory
reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), Grimm-Louis (2005),
Grimm-Lopes (2011), Kerstan-Weigand (2011), Grimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van
Proeyen-Zagermann (2006)
• Closed string axions ai from dim. red. of p-forms C(p) on M1,3 ×X6 (C(p) ∈RR-forms + NS 2-form in Type II)
ai ≡ (2π)−1∫
ΣiC(p), p − cycle Σi ⊂ X6, i ∈ 1, . . . , h11
h21 + 1
Kinetic terms for p-forms C(p) ∈ ; kinetic terms for ai
• Including D-branes wrapping p-cycle Σi :
? D-brane Chern-Simons terms ; anomal. coupling “ai Tr(G ∧ G)”
? Eucl. D-branes ; additional non-pert. effects (D-brane instantons)
• For a single D-brane wrapping p-cycle with ΩR(Σi ) 6= Σi
; ai acquires Stuckelberg charges under U(1)
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Introduction Mixing Axions and Large Decay Constants Open issues
Axions & String Theory: Example
reviews: Blumenhagen-Cvetic-Langacker-Shiu (’05); Blumenhagen-Kors-Lust-Stieberger (’06); Ibanez-Uranga (’12)
e.g. Type IIA D6-branes on CY3/ΩR ; Σi = Σi+ + Σi
−
∫Σi−
C(5) ∧ F 6= 0 ; Stuckelberg coupling for ai
T 6/ΩR with 4 ΩR-even 3-cycles Σi=0,1,2,3+︸ ︷︷ ︸
4 axions ai
and 4 ΩR-odd 3-cycles Σi=0,1,2,3−
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Introduction Mixing Axions and Large Decay Constants Open issues
Axions & String Theory: Example
reviews: Blumenhagen-Cvetic-Langacker-Shiu (’05); Blumenhagen-Kors-Lust-Stieberger (’06); Ibanez-Uranga (’12)
e.g. Type IIA D6-branes on CY3/ΩR ; Σi = Σi+ + Σi
−
∫Σi−
C(5) ∧ F 6= 0 ; Stuckelberg coupling for ai
T 6/ΩR with 4 ΩR-even 3-cycles Σi=0,1,2,3+︸ ︷︷ ︸
4 axions ai
and 4 ΩR-odd 3-cycles Σi=0,1,2,3−
ΩR
ΠaΠa = Σ0
+ − Σ3+︸ ︷︷ ︸+ Σ1
− − Σ2−︸ ︷︷ ︸
⇓ ⇓a0Fa ∧ Fa
−a3Fa ∧ Fa
(da1 − Aa)2
(da2 + Aa)2
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Introduction Mixing Axions and Large Decay Constants Open issues
An Effective Action...w/ Shiu-Ye 1503.01015, 1503.02965 [hep-th]
• String Theory compactifications 4d EFT with mixing axions
Seffaxion =
∫ 1
2
N∑i,j=1
Gij (dai − k i A) ∧ ?4(daj − k j A)−1
8π2
(N∑
i=1
ri ai
)Tr(G ∧ G) + Lgauge
• 2 types of kinetic mixing
(1) metric mixing: Gij is not diagonal(2) U(1) mixing: k i 6= 0 for some i ∈ 1, . . . ,N
gauged axions: ai → ai + k iχ, A→ A + dχ
• Tr(G ∧ G )-term associated to non-Abelian gauge group; collective periodicity:
∑Ni=1 ri a
i '∑N
i=1 ri ai + 2π
• axions couple to D-brane instantons; individual periodicity: ai → ai + 2πν i , ν i ∈ Znote: effective contributions of D-brane instantons to Seff
axion is model-dependent
see e.g. Ibanez-Uranga (2007, 2012), Blumenhagen-Cvetic-Kachru-Weigand (2009)
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Introduction Mixing Axions and Large Decay Constants Open issues
An Effective Action...w/ Shiu-Ye 1503.01015, 1503.02965 [hep-th]
• String Theory compactifications 4d EFT with mixing axions
Seffaxion =
∫ 1
2
N∑i,j=1
Gij (dai − k i A) ∧ ?4(daj − k j A)−1
8π2
(N∑
i=1
ri ai
)Tr(G ∧ G) + Lgauge
• 2 types of kinetic mixing
(1) metric mixing: Gij is not diagonal(2) U(1) mixing: k i 6= 0 for some i ∈ 1, . . . ,N
gauged axions: ai → ai + k iχ, A→ A + dχ
• Tr(G ∧ G )-term associated to non-Abelian gauge group; collective periodicity:
∑Ni=1 ri a
i '∑N
i=1 ri ai + 2π
• axions couple to D-brane instantons; individual periodicity: ai → ai + 2πν i , ν i ∈ Znote: effective contributions of D-brane instantons to Seff
axion is model-dependent
see e.g. Ibanez-Uranga (2007, 2012), Blumenhagen-Cvetic-Kachru-Weigand (2009)
![Page 15: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/15.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
An Effective Action...w/ Shiu-Ye 1503.01015, 1503.02965 [hep-th]
• String Theory compactifications 4d EFT with mixing axions
Seffaxion =
∫ 1
2
N∑i,j=1
Gij (dai − k i A) ∧ ?4(daj − k j A)−1
8π2
(N∑
i=1
ri ai
)Tr(G ∧ G) + Lgauge
• 2 types of kinetic mixing
(1) metric mixing: Gij is not diagonal(2) U(1) mixing: k i 6= 0 for some i ∈ 1, . . . ,N
gauged axions: ai → ai + k iχ, A→ A + dχ
• Tr(G ∧ G )-term associated to non-Abelian gauge group; collective periodicity:
∑Ni=1 ri a
i '∑N
i=1 ri ai + 2π
• axions couple to D-brane instantons; individual periodicity: ai → ai + 2πν i , ν i ∈ Znote: effective contributions of D-brane instantons to Seff
axion is model-dependent
see e.g. Ibanez-Uranga (2007, 2012), Blumenhagen-Cvetic-Kachru-Weigand (2009)
![Page 16: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/16.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
An Effective Action...w/ Shiu-Ye 1503.01015, 1503.02965 [hep-th]
• String Theory compactifications 4d EFT with mixing axions
Seffaxion =
∫ 1
2
N∑i,j=1
Gij (dai − k i A) ∧ ?4(daj − k j A)−1
8π2
(N∑
i=1
ri ai
)Tr(G ∧ G) + Lgauge
• 2 types of kinetic mixing
(1) metric mixing: Gij is not diagonal(2) U(1) mixing: k i 6= 0 for some i ∈ 1, . . . ,N
gauged axions: ai → ai + k iχ, A→ A + dχ
• Tr(G ∧ G )-term associated to non-Abelian gauge group; collective periodicity:
∑Ni=1 ri a
i '∑N
i=1 ri ai + 2π
• axions couple to D-brane instantons; individual periodicity: ai → ai + 2πν i , ν i ∈ Znote: effective contributions of D-brane instantons to Seff
axion is model-dependent
see e.g. Ibanez-Uranga (2007, 2012), Blumenhagen-Cvetic-Kachru-Weigand (2009)
![Page 17: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/17.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
...for Mixing Axions
w/ Shiu-Ye 1503.01015, 1503.02965 [hep-th]
To determine axion decay constantsTo figure out axions eaten by AU(1)
Diagonalisekinetic terms
=⇒
eigenbasisfor kinetic terms
6= eigenbasisfor potentials
⇓axionic directions
with large fa?
Note: different from N-flation Dimopoulos-Kachru-McGreevy-Wacker (2005),KNP-alignment with large N Choi-Kim-Yung (2014),Kinematic alignment (with RMT) Bachlechner-Long-McAllister (2014/15), Junghans (2015)
for N axions: feff ∼ Npf with p ≥ 1/2
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Introduction Mixing Axions and Large Decay Constants Open issues
...for Mixing Axions
w/ Shiu-Ye 1503.01015, 1503.02965 [hep-th]
To determine axion decay constantsTo figure out axions eaten by AU(1)
Diagonalisekinetic terms
=⇒
eigenbasisfor kinetic terms
6= eigenbasisfor potentials
⇓axionic directions
with large fa?
Note: different from N-flation Dimopoulos-Kachru-McGreevy-Wacker (2005),KNP-alignment with large N Choi-Kim-Yung (2014),Kinematic alignment (with RMT) Bachlechner-Long-McAllister (2014/15), Junghans (2015)
for N axions: feff ∼ Npf with p ≥ 1/2
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Introduction Mixing Axions and Large Decay Constants Open issues
2 Mixing Axions
• minimal set-up: 2 axions + 1 U(1) + 1 Non-Abelian gauge group
• 1 axion eaten by U(1) gauge boson, ⊥ axion ξ with decay constant:
fξ =
√λ+λ−Mst
cos θ2
(λ+k+r2 + λ−k−r1) + sin θ2
(λ−k−r2 − λ+k+r1)
with λ± eigenvalues of Gij and Mst ≡√λ+(k+)2 + λ−(k−)2
cos θ =G11 − G22
λ+ − λ−, sin θ =
2G12
λ+ − λ−,
(k+
k−
)=
(cos θ
2sin θ
2sin θ
2− cos θ
2
)(k1
k2
)
• Contour plot representation of fξ (in units√G11)
![Page 20: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/20.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
2 Mixing Axions
• minimal set-up: 2 axions + 1 U(1) + 1 Non-Abelian gauge group
• 1 axion eaten by U(1) gauge boson, ⊥ axion ξ with decay constant:
fξ =
√λ+λ−Mst
cos θ2
(λ+k+r2 + λ−k−r1) + sin θ2
(λ−k−r2 − λ+k+r1)
with λ± eigenvalues of Gij and Mst ≡√λ+(k+)2 + λ−(k−)2
cos θ =G11 − G22
λ+ − λ−, sin θ =
2G12
λ+ − λ−,
(k+
k−
)=
(cos θ
2sin θ
2sin θ
2− cos θ
2
)(k1
k2
)
• Contour plot representation of fξ (in units√G11)
![Page 21: Adventures with Stringy AxionsIntroductionMixing Axions and Large Decay ConstantsOpen issues In ation and Axions BICEP2 (2014) )illusion of large eld in ation Lyth (1996): tensor pert](https://reader036.vdocuments.pub/reader036/viewer/2022070220/6130d90b1ecc515869445bd6/html5/thumbnails/21.jpg)
Introduction Mixing Axions and Large Decay Constants Open issues
2 Mixing Axions
• minimal set-up: 2 axions + 1 U(1) + 1 Non-Abelian gauge group
• 1 axion eaten by U(1) gauge boson, ⊥ axion ξ with decay constant:
fξ =
√λ+λ−Mst
cos θ2
(λ+k+r2 + λ−k−r1) + sin θ2
(λ−k−r2 − λ+k+r1)
with λ± eigenvalues of Gij and Mst ≡√λ+(k+)2 + λ−(k−)2
cos θ =G11 − G22
λ+ − λ−, sin θ =
2G12
λ+ − λ−,
(k+
k−
)=
(cos θ
2sin θ
2sin θ
2− cos θ
2
)(k1
k2
)
• Contour plot representation of fξ (in units√G11)
√G22G11
θ
2k1 = k2 = 2r1 = 2r2
√G22G11
θ
k1 = 2k2 = r1 = 2r2
√G22G11
θ
k1 = −2k2 = r1 = 2r2
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Introduction Mixing Axions and Large Decay Constants Open issues
Consistency Conditions• U(1) gauge invariance: A→ A + dχ, ζ → ζ + kχ
Ssub =
∫ f 22
2
(dζ − kA
)∧ ?4
(dζ − kA
)−
1
g21
F ∧ ?4F −1
8π2ζ Tr( G ∧ G)︸ ︷︷ ︸
not U(1) invariant
U(1) ζ
G
G
• Integrating out massive U(1) boson 1 axion ξ + 1 non-Abelian gauge group + chiral fermions
S =
∫1
2dξ ∧ ?4dξ −
1
8π2
ξ
fξTr(G ∧ G)−
Cf 22
Jψ ∧ ?4Jψ︸ ︷︷ ︸4−fermion
+Lψ
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Introduction Mixing Axions and Large Decay Constants Open issues
Consistency Conditions• U(1) gauge invariance: A→ A + dχ, ζ → ζ + kχ
Ssub =
∫ f 22
2
(dζ − kA
)∧ ?4
(dζ − kA
)−
1
g21
F ∧ ?4F −1
8π2ζ Tr( G ∧ G)︸ ︷︷ ︸
not U(1) invariant
U(1) ζ
G
G
+
“reversed” GS mechanism
+ chiral fermions ψ
U(1)ψ
ψ
ψG
G
• Integrating out massive U(1) boson 1 axion ξ + 1 non-Abelian gauge group + chiral fermions
S =
∫1
2dξ ∧ ?4dξ −
1
8π2
ξ
fξTr(G ∧ G)−
Cf 22
Jψ ∧ ?4Jψ︸ ︷︷ ︸4−fermion
+Lψ
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Introduction Mixing Axions and Large Decay Constants Open issues
Consistency Conditions• U(1) gauge invariance: A→ A + dχ, ζ → ζ + kχ
Ssub =
∫ f 22
2
(dζ − kA
)∧ ?4
(dζ − kA
)−
1
g21
F ∧ ?4F −1
8π2ζ Tr( G ∧ G)︸ ︷︷ ︸
not U(1) invariant
U(1) ζ
G
G
+
“reversed” GS mechanism
+ chiral fermions ψ
U(1)ψ
ψ
ψG
G
+
+ generalised CS-terms
for non-symm Aanomaly
U(1)
G
G
Aldazabel-Ibanez-Uranga (’03), Anastasopoulos et al (’06)
De Rydt-Rosseel-Schmidt-Van Proeyen-Zagerman (’07)
= 0
• Integrating out massive U(1) boson 1 axion ξ + 1 non-Abelian gauge group + chiral fermions
S =
∫1
2dξ ∧ ?4dξ −
1
8π2
ξ
fξTr(G ∧ G)−
Cf 22
Jψ ∧ ?4Jψ︸ ︷︷ ︸4−fermion
+Lψ
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Introduction Mixing Axions and Large Decay Constants Open issues
Consistency Conditions• U(1) gauge invariance: A→ A + dχ, ζ → ζ + kχ
Ssub =
∫ f 22
2
(dζ − kA
)∧ ?4
(dζ − kA
)−
1
g21
F ∧ ?4F −1
8π2ζ Tr( G ∧ G)︸ ︷︷ ︸
not U(1) invariant
• Integrating out massive U(1) boson 1 axion ξ + 1 non-Abelian gauge group + chiral fermions
S =
∫1
2dξ ∧ ?4dξ −
1
8π2
ξ
fξTr(G ∧ G)−
Cf 22
Jψ ∧ ?4Jψ︸ ︷︷ ︸4−fermion
+Lψ
ψ
ψU(1)
ψ
ψ
=⇒ + global U(1)anom
ψ
ψ
ψ
ψ
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Introduction Mixing Axions and Large Decay Constants Open issues
Inflationary potential
@ strong coupling: fermion condensate 〈ψψ〉 + gluon condensate 〈GG〉
? global U(1)anom ∂µJµU(1)
=1
16π2Tr(Gµν Gµν)︸ ︷︷ ︸
gluon condensate
→ mass-term for phase of ψψ∼ η(′)-meson
? 4-Fermion coupling mass-terms for axion ξsee e.g. reviews ’t Hooft (1986), Peccei (2006), Svrcek-Witten (2006)
⇒ Effective potential
Veff (ξ, η) = Vfermion(ξ) + Vgluon(η, ξ)
In vacuum of Veff mηmξ∼ fξ
fη 1
Upon integrating out η-meson
Veff (ξ) = Λ4 cos
(ξ
fξ− π
)Λ4 ∼
(〈ψψ〉
f2
)2
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Introduction Mixing Axions and Large Decay Constants Open issues
Conclusions
• Enhanced fξ for mixing axions charged under U(1)
if ∃ isotropy relations among contin. moduli λ− & λ+ (white regions)
• Vinf : from non-perturbative effects of 1 Non-Abelian gauge group
• Realisations in String Theory:
IIA w/ inters. D6-branesIIB w/ inters. magnetised D7-branes
−→ more suitable examples are desired
Open issues
Validity of eff. description requires moduli stabilisation??Conlon (2006), Choi-Jeong (2006), Hristov (2008), Higaki-Kobayashi (2011),
Cicoli-Dutta-Maharana (2014)
No-go theorems for multiple axions??Brown-Cottrell-Shiu-Soler(2015), Montero-Uranga-Valenzuela (2015), Junghans (2015),
Rudelius (2014/15), Hebecker-Mangat-Rompineve-Witkowski (2015), Bachlechner-Long-McAllister (2015)
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Introduction Mixing Axions and Large Decay Constants Open issues
Conclusions
• Enhanced fξ for mixing axions charged under U(1)
if ∃ isotropy relations among contin. moduli λ− & λ+ (white regions)
• Vinf : from non-perturbative effects of 1 Non-Abelian gauge group
• Realisations in String Theory:
IIA w/ inters. D6-branesIIB w/ inters. magnetised D7-branes
−→ more suitable examples are desired
Open issues
Validity of eff. description requires moduli stabilisation??Conlon (2006), Choi-Jeong (2006), Hristov (2008), Higaki-Kobayashi (2011),
Cicoli-Dutta-Maharana (2014)
No-go theorems for multiple axions??Brown-Cottrell-Shiu-Soler(2015), Montero-Uranga-Valenzuela (2015), Junghans (2015),
Rudelius (2014/15), Hebecker-Mangat-Rompineve-Witkowski (2015), Bachlechner-Long-McAllister (2015)
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Introduction Mixing Axions and Large Decay Constants Open issues
Obrigado Muchas gracias
Eskerrik asko
Moitas grazas
Moltes gracies
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Introduction Mixing Axions and Large Decay Constants Open issues
WGC in a Perpetuum Mobile
• WGC (weak form) states that ∃ state with(
MQ
)≤ MPl
; conjectured generalisation for 0-forms with Sinst ≤ MPlf
Arkhani-Hamed-Motl-Nicolis-Vafa (2006)
• Consistent compactifications for
M-theoryS,T←→ Type IIB
M1,2 × S1M × S1 × X6 M1,2 × S1 × X6
; Constraints on effective axion decay constant by applying WGC on 5dim BHin M-theoryBrown-Cottrell-Shiu-Soler (2015)
• Generalisations of WGC → Lattice WGC (stronger form!)Heidenreich-Reece-Rudelius (2014/15)
• BUT in our simple model: 2 axions + 1 U(1) + 1 gauge instanton
Does it satisfy Lattice WGC?Are higher harmonics are still troublesome for ⊥ axion?Montero-Uranga-Valenzuela (2015)
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Introduction Mixing Axions and Large Decay Constants Open issues
Inflation and Axions• Single field models discriminated by Lyth-bound: Lyth (1996)
∆φ
MPl= O(1)
( r
0.01
)1/2 r < 0.01 small field inflation (∆φ < MPl )r > 0.01 large field inflation (∆φ > MPl )
; measurable tensor perb. ∆2T (k) for r ≡ ∆2
T (k)
∆2S (k)
> 0.01
• η-problem: sensitivity to dim 6 operators ⇒ |∆η| O(1) slow roll• Shift symmetry of axions forbids such corrections, while inflaton
potential follows from non-perturb. effects
V (a) ∼ Λ4[1− cos
a
f
]⇒ natural inflation Freese-Frieman-Olinto (1990) with f > MPl for slow roll
• String Theory: reviews: Baumann (2009), Baumann-McAllister (2009,2014), Westphal (2014), . . .
? UV complete theory with plethora of axions? local origin of shift symmetries Banks-Dixon (1988),
Kallosh-Linde-Linde-Susskind (1995), Banks-Seiberg (2010), . . .
But ∃ no-go theorems and arguments forbidding f > MPl (1 axion)Banks-Dine-Fox-Gorbatov (2003), Svrcek-Witten (2006), Arkhani-Hamed-Motl-Nicolis-Vafa (2006)
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Introduction Mixing Axions and Large Decay Constants Open issues
Main question
Apart from 3 traditional multiple axion scenarios
• N-flation Dimopoulos-Kachru-McGreevy-Wacker (2005)
• aligned natural inflation (KNP) Kim-Nilles-Peloso (2004), Kappl-Krippendorf-Nilles
(2014), Choi-Kim-Yun (2014), Ben-Dayan-Pedro-Westphal (2014), Long-McAllister-McGuirk (2014)
• axion monodromy Silverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008),
Kaloper-(Lawrence)-(Sorbo) (2008/11/14), Marchesano-Shiu-Uranga (2014),
Franco-Galloni-Retolaza-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Hebecker-Kraus-Witkowski (2014), Blumenhagen-Herschmann-Plauschinn (2014),
Retolaza-Uranga-Westphal (2015)
∃ other scenarios to obtain f > MPl in string theory?
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Introduction Mixing Axions and Large Decay Constants Open issues
Some aspects of Inflationreviews: Baumann (2009); Baumann-McAllister (2009,2014); Westphal (2014); . . .
• Inflation = cure for horizon problem and flatness problem
• Inflation = explanation for fluctuations in nearly scale-invariant,nearly Gaussian CMB
• typical model: slow-roll single scalar field with potential V satisfying
ε ≡M2
Pl
2
(V ′
V
)2
1, |η| ≡∣∣∣∣M2
Pl
V ′′
V
∣∣∣∣ 1 during inflation
V (φ)
φ∆φ
δφ
∆φ: distance traveled during inflation
• QM fluctuations δφ during inflation
⇒
scalar pert. ∆2S (k)
tensor pert. ∆2T (k)
∼ 10−9 (WMAP+PLANCK)
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Introduction Mixing Axions and Large Decay Constants Open issues
Some aspects of Inflation• spectral index ns : deviation from scale-invariance
∆S (k) = AS
(k
k?
)ns−1
k? : reference scale
• tensor-to-scalar ratio r : r ≡ ∆2T (k)
∆2S (k)
sets the inflation scale
V 1/4 ∼( r
0.01
)1/41016 GeV
• Lyth bound: relates field displacements ∆φ to r during inflation
∆φ
MPl= O(1)
( r
0.01
)1/2
r < 0.01 small field inflation (∆φ < MPl )r > 0.01 large field inflation (∆φ > MPl )
• (ns , r) are related to slow-roll parameters (ε, η):
ns − 1 = 2η − 6ε r = 16ε
⇒ (ns , r) measurements give direct info about potential V
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Introduction Mixing Axions and Large Decay Constants Open issues
Some Considerations about Axions
• axions = CP-odd real scalars with a continuous shift symmetry:
a→ a + ε, ε ∈ R
• shift symmetry is broken to a discrete symmetry by non-perturbativeeffects (gauge instantons, D-brane instantons, etc.):
a→ a + 2πn, n ∈ Z symmetry constrains axionic couplings:
S =
∫f 2a
2da ∧ ?4da− Λ4 [1± cos(a)] ?4 1
fa: axion decay constants (coupling strength of axion to other matter)
• natural inflation: axion as inflaton candidate Freese-Frieman-Olinto (1990)
V (a)
a2πfa
ε =M2
Pl2f 2
a
(sin(a/fa)
1±cos(a/fa)
)2 1
η =M2
Plf 2a
∣∣∣ cos(a/fa)1±cos(a/fa)
∣∣∣ 1
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Introduction Mixing Axions and Large Decay Constants Open issues
“Lifting” the flat direction
(1) Monodromy effects: shift symmetry is softly broken
• F-term (torsional cycles, fluxes) −→ V (ξ) ∼ ξp≥2
Marchesano-Shiu-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Blumenhagen-Herschmann-Plauschinn (2014),
• D-term (D-branes) −→ V (ξ) =√
L4 + ξ2 ∼ ξSilverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008)
Kaloper-Lawrence-Sorbo (2008-2014)
(2) Alignment effects: 2 non-Abelian gauge groupsKim-Nilles-Peloso (2004), Kappl-Krippendorf-Nilles (2014); Choi-Kim-Yun (2014)
V effaxion = Λ4
1
[1− cos
(a−
f1+
a+
g1
)]+ Λ4
2
[1− cos
(a−
f2+
a+
g2
)]see also BenDayan-Pedro-Westphal (2014), Long-McAllister-McGuirk (2014)
(3) U(1) gauge symmetry:
• 1 axionic direction eaten by U(1) boson (Stuckelberg mechanism)
• orthogonal direction acquires mass due to non-perturbative effect
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Introduction Mixing Axions and Large Decay Constants Open issues
“Lifting” the flat direction
(1) Monodromy effects: shift symmetry is softly broken
• F-term (torsional cycles, fluxes) −→ V (ξ) ∼ ξp≥2
Marchesano-Shiu-Uranga (2014), McAllister-Silverstein-Westphal-Wrase (2014),
Blumenhagen-Herschmann-Plauschinn (2014),
• D-term (D-branes) −→ V (ξ) =√
L4 + ξ2 ∼ ξSilverstein-Westphal (2008), McAllister-Silverstein-Westphal (2008)
Kaloper-Lawrence-Sorbo (2008-2014)
(2) Alignment effects: 2 non-Abelian gauge groupsKim-Nilles-Peloso (2004), Kappl-Krippendorf-Nilles (2014); Choi-Kim-Yun (2014)
V effaxion = Λ4
1
[1− cos
(a−
f1+
a+
g1
)]+ Λ4
2
[1− cos
(a−
f2+
a+
g2
)]see also BenDayan-Pedro-Westphal (2014), Long-McAllister-McGuirk (2014)
(3) U(1) gauge symmetry:
• 1 axionic direction eaten by U(1) boson (Stuckelberg mechanism)
• orthogonal direction acquires mass due to non-perturbative effect
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Introduction Mixing Axions and Large Decay Constants Open issues
Aligned natural inflation• Consider 2 strongly coupled non-Abelian gauge groups:
V effaxion = Λ4
1
[1− cos
(a−
f1+
a+
g1
)]+ Λ4
2
[1− cos
(a−
f2+
a+
g2
)]with decay constants
f1 =
√λ−
|r1 sin θ2− r2 cos θ
2|
g1 =
√λ+
|r1 cos θ2
+ r2 sin θ2|
f2 =
√λ−
|s1 sin θ2− s2 cos θ
2|
g2 =
√λ+
|s1 cos θ2
+ s2 sin θ2|
• Perfect alignment:
f1
g1=
f2
g2⇒
∣∣∣∣∣ r1 cos θ2
+ r2 sin θ2
r1 sin θ2− r2 cos θ
2
∣∣∣∣∣ =
∣∣∣∣∣ s1 cos θ2
+ s2 sin θ2
s1 sin θ2− s2 cos θ
2
∣∣∣∣∣• Deviation from perfect alignment
αdev ≡ g2 −f2
f1g1 =
√λ+ (s1r2 − r1s2)(
s21−s2
22
sin θ − s1s2 cos θ
)(r1 cos θ
2+ r2 sin θ
2
)
• Continuous parameter θ allows for αdev ≈ 0.009√λ+ with ri , si ∼ O(1− 10)
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Introduction Mixing Axions and Large Decay Constants Open issues
U(1) gauge invariance• U(1) gauge invariance: A→ A + dχ, a′2 → a′2 + kχ
Ssub =
∫ f 22
2
(da′2 − kA
)∧ ?4
(da′2 − kA
)−
1
g21
F ∧ ?4F −1
8π2a′2 Tr( G ∧ G)︸ ︷︷ ︸
not U(1) invariant
; requires presence of chiral fermions
“reversed” GS mechanism ⇒ δSmixanom = −
∫1
8π2AmixχTr( G ∧ G)
• If anomaly also contains non-symmetric contributions; Generalised Chern-Simons terms Aldazabel-Ibanez-Uranga (2003), Anastasopoulos et al (2006)
De Rydt-Rosseel-Schmidt-Van Proeyen-Zagerman (2007)
SGCSsub =
∫1
8π2AGCSA ∧ Ω,
• Full U(1) gauge invariance: k +Amix +AGCS = 0
• + 3 additional constraints from non-Abelian and mixedAbelian/non-Abelian anomaly cancelation
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Introduction Mixing Axions and Large Decay Constants Open issues
Anomalies
• Spectrum of chiral fermionsSU(N) U(1)
ψiL R i
1 qiL
ψiR R i
2 qiR
• mixed anomaly coefficient:
Amix =∑
i
[Tr(qi
LTR i
1a ,T
R i1
b )− Tr(qiRT
Ri2
a ,TR
i2
b )]
• mixed anomaly: AGCS −Amix = 0
• cubic U(1) anomaly: AU(1)3=∑
i
[(qi
L)3 − (qiR )3]
= 0
• cubic SU(N) anomaly:
ASU(N)3=∑
i
[Tr(T
R i1
a TR i
1b ,T
R i1
c )− Tr(TR
i2
a TR
i2
b ,TR
i2
c )]
= 0
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Introduction Mixing Axions and Large Decay Constants Open issues
Integrating out
• Potential for a′1? → integrating out A (+ a′2)
Step 1: e.o.m for massive A in unitary gauge
−1
g21
d(?4dA) + (fa2k2)2 ?4 A = −
AGCS
8π2Ω− ?4Jψ
Step 2: Deduce Lorenz-gauge condition
(fa2 k2)2d(?4A) = −AGCS +Amix
Amixd(?4Jψ)
Step 3: Re-insert relation between A and Jψ
S =
∫f 21
2da′1 ∧ ?4da′1 −
1
8π2a′1Tr(G ∧ G)−
Cf 22
Jψ ∧ ?4Jψ︸ ︷︷ ︸4−fermion
• Potential for a′1? → integrating out non-Abelian gauge group +chiral fermions
S =
∫f 21
2da′1 ∧ ?4da′1 − Λ4
[1− cos
(a′1)]?4 1
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Introduction Mixing Axions and Large Decay Constants Open issues
Axions from String Theory
Dimensional reduction of String Theory on M1,3 ×X6
⇒ plethora of axions see e.g. Witten (1984), Banks-Dine-Fox-Gorbatov (2003), Svrcek-Witten (2006)
(1) Closed String Axions
• 1 Model-independent: NS 2-form B2 along M1,3
• h11 + h21 Model-dependent: NS 2-form B2
RR-forms Cpalong X6
(2) Open String Axions
• Wilson-line: reduction of (D-brane) gauge fieldsee e.g. ArkaniHamed-Cheng-Creminelli-Randal, Marchesano-Shiu-Uranga (2014)
• Field-type: phase of C scalar field in N = 1 chiral multiplet @intersection of 2 D-branes
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Introduction Mixing Axions and Large Decay Constants Open issues
String Theory embedding I
Grimm-Louis (2005), Grimm-Lopes (2011), Kerstan-Weigand (2011)
• Type IIA on M1,3 ×X6 with D6-branesM1,3 maximally symmetric 4-dim spacetimeX6 admits a symplectic basis (αi , β
j ) for H3(X6):∫X6αi ∧ βj = `6
s δij
e.g. X6 = CY3/ΩR
• axions emerge from reduction of RR-form C3& charges under (D-brane) U(1) from reduction of RR-form C5:
C3 =1
2π
b3∑i=1
ξi (x)αi (y) + . . . C5 =1
2π
b3∑i=1
D(2)i ∧ βi + . . .
• Reduction of bulk action −→ kinetic terms
SbulkR 3 −
π
2`8s
∫dCi∧?4dCi=3,5 −→ Skin = −
1
4`2s
∫dξi∧?4dξjKij +dD(2)i∧?4dD(2)j
Kij
with Kij = 12π`6
s
∫X6αi ∧ ?6αj and Kij = K−1
ij
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Introduction Mixing Axions and Large Decay Constants Open issues
String Theory embedding II
• D6-brane wraps M1,3 ×∆3, w.r.t. de Rahm-dual basis (γi , δj )
∆3 =
b3/2∑i=1
(r iγi + piδi )
∫γj
αi = `3s δi
j =
∫δiβj
• Reduction of D-brane action −→ anomalous coupling + U(1)charges
SD6CS 3
∫D6
1
4π`3s
C3∧F 2+1
`5s
C5∧F −→1
8π2
b3/2∑i=1
r i∫M1,3
ξi F∧F +1
2π`2s
b3/2∑j=1
pj
∫M1,3
D(2)j∧dA
• D(2)i is Hodge-dual to ξi ; dualisation in favour of ξi :
Saxion = −1
2`2s
∫M1,3
[1
2
(dξi −
pi
πA)∧ ?4
(dξj −
pj
πA)Kij
]+
1
8π2
∑i
r i∫M1,3
ξi F ∧ F
• similar reasoning for C4-axions in type IIB with D7-branesGrimm-Louis (2004), Jockers-Louis (2005), Haack-Krefl-Lust-Van Proeyen-Zagermann (2006)